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i) Given that the complex numbers Z, and Z, where Z, = (1+i), z, express Z 2+i 3-i and Z, in the form a + ib, where a, b are real. ii) Solve the simultaneous equations Z3 + Z4 = 6, 2Z3 - 2iZ4 = 8+ 3i, expressing your answer in the form a + ib, where a and b are real numbers.

i) Given that the complex numbers Z, and Z, where Z, = (1+i), z, express Z 2+i 3-i and Z, in the form a + ib, where a, b are real. ii) Solve the simultaneous equations Z3 + Z4 = 6, 2Z3 - 2iZ4 = 8+ 3i, expressing your answer in the form a + ib, where a and b are real numbers.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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otherwise, find the roots of the equatiön 3. i) Given that the complex numbers z,and z, are the roots of the equation: x² - 4i – 3 = 0. Express z,and z, in the form a + bi where a and b are real. ii) Indicate by shaded areas on separate Argand diagrams the regions defined by: a) /Z + 3/<1_b) /Z+4/>/Z+1/ c) {"< ArgZ < "}n {Im Z < 4} %3D 3 2 T - isin -) in the form a + bi, where . :3 4. a) Given that z = 2- 3i, express i'z, and Z/Z* and Z (cos- 4 4 a and b are real. Z+1 b) Given that Fi, find z in the forms a + bi, where a and b are real. Z-1 c) Given that the complex number 2i represents the point A on the Argand diagram and that the point B represents the image of A when reflected in the line y = 2x, find the complex number which represents the point B. 5. i) Find two complex numbers Z, and Z, which satisfy the simultaneous equations Z, + Z2 = -i, Z, – iZ2= -4 +i ii) Given that Z= 1 + iv3 express Z* in the form | a) a+ bi where a and b are real. b) r(cosq + isinq), where r> 0 and - 1<q<I iii) By shading in three separate Argand diagrams show the regions in which the points representing z can lie when (a) Imz< 2 b) /Z-2i/ < 2. c) Z--2i/</Z - 2/. Shade in another Argand diagram the region in whichz can lie when ail the three inequalities apply 6. i) Given thati the complex numbers Z, and Z2 where Z, = (1+i), Z2 1 1 express Z 2+i 3-i and Za in the form a + ib, where a, b are real. ii) Solve the simultaneous equations Z; + Z4 = 6, 2Z; - 2iZ4 = 8+ 3i, expressing your %3D answer in the form a + ib, where a and b are real numbers. iii) A regular hexagon ABC DEF is drawn in the Argand diagram so that it centre is at the origin and the two adjacent vertices A and B are at the points represented by the complex numbers ZA and ZB = 1 and ZB = ½ + i½v3. Find, in the form a + bi where a and b are real numbers, the complex numbers which represent the other four vertices. 7. Given that /2Z - t/= /Z-3i/, show that the locus of the complex number Z is a circle, giving %3D the radius and the coordinates of the centre. 8. a) Find in the form a + bi, where a and b are real, the values of Z for which 3 i) /z/=3 and arg Z ii) + =1. = 1. 1-2i 3 b) Find the roots of the equation: Z* +6Z“ + 25 = 0. Represent these roots as points on the Arganddiagram., 9. i) Express each of the complex numbers Z= (2+3i)(1- 2i), Z2 = (3 + 5i)/2 - i in the form a + %3D %3D Z2 ib where a and b are real numbers. Find arg Z2 and arg gi.ng your answer in degrees Z1 correct to one decimal place. 76
Transcribed Image Text:otherwise, find the roots of the equatiön 3. i) Given that the complex numbers z,and z, are the roots of the equation: x² - 4i – 3 = 0. Express z,and z, in the form a + bi where a and b are real. ii) Indicate by shaded areas on separate Argand diagrams the regions defined by: a) /Z + 3/<1_b) /Z+4/>/Z+1/ c) {"< ArgZ < "}n {Im Z < 4} %3D 3 2 T - isin -) in the form a + bi, where . :3 4. a) Given that z = 2- 3i, express i'z, and Z/Z* and Z (cos- 4 4 a and b are real. Z+1 b) Given that Fi, find z in the forms a + bi, where a and b are real. Z-1 c) Given that the complex number 2i represents the point A on the Argand diagram and that the point B represents the image of A when reflected in the line y = 2x, find the complex number which represents the point B. 5. i) Find two complex numbers Z, and Z, which satisfy the simultaneous equations Z, + Z2 = -i, Z, – iZ2= -4 +i ii) Given that Z= 1 + iv3 express Z* in the form | a) a+ bi where a and b are real. b) r(cosq + isinq), where r> 0 and - 1<q<I iii) By shading in three separate Argand diagrams show the regions in which the points representing z can lie when (a) Imz< 2 b) /Z-2i/ < 2. c) Z--2i/</Z - 2/. Shade in another Argand diagram the region in whichz can lie when ail the three inequalities apply 6. i) Given thati the complex numbers Z, and Z2 where Z, = (1+i), Z2 1 1 express Z 2+i 3-i and Za in the form a + ib, where a, b are real. ii) Solve the simultaneous equations Z; + Z4 = 6, 2Z; - 2iZ4 = 8+ 3i, expressing your %3D answer in the form a + ib, where a and b are real numbers. iii) A regular hexagon ABC DEF is drawn in the Argand diagram so that it centre is at the origin and the two adjacent vertices A and B are at the points represented by the complex numbers ZA and ZB = 1 and ZB = ½ + i½v3. Find, in the form a + bi where a and b are real numbers, the complex numbers which represent the other four vertices. 7. Given that /2Z - t/= /Z-3i/, show that the locus of the complex number Z is a circle, giving %3D the radius and the coordinates of the centre. 8. a) Find in the form a + bi, where a and b are real, the values of Z for which 3 i) /z/=3 and arg Z ii) + =1. = 1. 1-2i 3 b) Find the roots of the equation: Z* +6Z“ + 25 = 0. Represent these roots as points on the Arganddiagram., 9. i) Express each of the complex numbers Z= (2+3i)(1- 2i), Z2 = (3 + 5i)/2 - i in the form a + %3D %3D Z2 ib where a and b are real numbers. Find arg Z2 and arg gi.ng your answer in degrees Z1 correct to one decimal place. 76
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